1
Developing an MOR/ML Dynamic Programming Program
At this point, we have modeled a variety of dynamic programming models and solved
them both by hand and by use of the MOR/DP support package.
It is now time to
consider the details of the computational mechanics of dynamic programming.
Therefore, we will revisit some of these example problems and develop programs to
implement their solution in MOR/ML.
There are two levels of accomplishment possible.
First, being able to develop a program to solve a specific fixed length problem requires a
certain amount of knowledge about the process.
Then to actually generalize the program
to an
n
-dimensional problem based on the problem data requires a bit more programming
and MOR/ML manipulation skills.
We begin by restating the limited-inventory
production-planning example problem of the introduction.
Problem.
The demand for a product over the next four time periods is 2, 3, 4, and 2
units.
The cost of placing an order is $15, independent of the number of units ordered.
The individual item cost is $100, and the holding is $2 per unit per period.
Due to the
company accounting system, holding costs are charged on the inventory units at the
beginning of a period; that is, on entering inventory.
A maximum of four units can be
held from period to period.
The company orders from a local supplier, so we can
consider that units ordered in a period are available for use in the same period.
However,
due to the limited production capacity of the supplier, orders are limited to once a period
with a maximum of five units.
We want the optimal ordering policy that meets the demands at minimal cost for
the four time periods.
The company currently has no items in inventory and has no
requirements for a specified number in inventory at the end of the planning horizon.
Since demands must be met, ordering for future demand is the only viable option for
decreasing costs.
Solution.
We have a four period decision problem with production quantities
x
i
to be
determined.
The inventory at the beginning of each period is represented by the variable
s
i
, with the initial inventory,
s
1
, set to zero.
The period demand for units is defined as
d
i
.
The relationships between these variables are:
s
i
+
1
=
s
i
+
x
i
−
d
i
∈
0,1,2,3,4
{
}
,

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